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POPULATION OF GROUND-STATE
ROTATIONAL BANDS OF
SUPERHEAVY NUCLEI PRODUCED
IN COMPLETE FUSION
REACTIONS
A.S. Zubov, V.V. Sargsyan, G.G. Adamian, N.V.Antonenko
Introduction/Motivation



The existence of heaviest nuclei with Z > 100 is mainly determined
by their shell structure.
Although different models provide close binding energies for
nuclei with Z < 114, their predictions for the spins of the ground
states, quasiparticle spectra, and moments of inertia are rather
different.
The limits of stability of SH nuclei in spin and excitation energy are
governed by the fission barrier, which is mostly defined by the shell
component.
Introduction/Motivation
 We extend the statistical approach to describe the population of yrast
rotational states in the residual superheavy nuclei and the production of
evaporation residues in the complete fusion reactions
206,208Pb(48Ca,2n)252,254No and 204Hg(48Ca,2n)250Fm, taking into
consideration the survival of compound nuclei against fission at different
angular momenta.
Our aim is the study of damping of the shell effects with the angular
momentum for shell-stabilized nuclei.
 We try to answer the question of how the angular momentum
dependence of the survival of the compound nucleus against fission is
important for the production of new superheavy elements.
Model
Our model treats the formation of the yrast residual nucleus as a three-step
process.
Capture
Fusion
Yrast residual nucleus
neutron
neutron
statistical
rotating dinuclear system
is formed in the entrance
channel by the transition
of the colliding nuclei
over
the
entrance
(Coulomb) barrier.
dinuclear system evolves
into
the
compound
nucleus in the mass
(charge)
asymmetry
coordinate
γ
excited rotating compound
nucleus transforms to the
yrast residual nucleus by the
emission of the neutrons and
statistical γ quanta
Model
 The population of yrast rotational state L+ with spin L
and positive parity depends on the survival of the
compound nucleus against fission in the xn
evaporation channel.
 The residual excitation energy E∗ after the emission of
neutrons should be less than the fission barrier and
quite small to be taken away by k statistical dipole γ
quanta.
Model
Тhe population cross section of the state L+
pop
ECN* , J , E* 
 pop Ec.m. , L    cap Ec.m. , J  Pcap Ec.m. , J Wsur
k
J  L  k l g  x ln
---- the initial angular momentum of the compound
nucleus
The average angular momentum carried out by neutrons and E1
statistical γ quanta was taken as ln = 0.5 and lg = 1, respectively.
*
J   Ec.m.  Q   2 J J  1 2
ECN
---- thermal excitation energy of the
produced nucleus
Model
Тhe neutron carries out from a nucleus the average energy 2T (by
assuming the Maxwellian form of neutron spectrum) and average
angular momentum 0.5.
*
J ,
E1*  ECN
1. step
2. step
….
*
J  a
T  ECN
E2*  E1*  2T ,
--
J1  J
J 2  J  0.5
nuclear temperature
Model
The survival probability in the case of the evaporation of x neutrons and k γ
quanta
pop
sur
W
Pxn k
E
*
CN

, J , E  Pxn k
*


n Ei* , J i
E , J , E 
*

E
i 1
f
i , Ji

*
CN
*

x


---- the probability of the realization of an xnkγ channel
The relative intensity of E2 multipolarity transitions between the yrast
rotational states L+ and (L−2)+

I L  L  2  

Lmax
2
xL2
Lmax
x 0
2
 pop Ec.m. ,2 x 
 pop Ec.m. ,2 x 
Evaporation residue
Partial evaporation residue cross section
 ER Ec.m. , J    cap Ec.m. , J  PCN Ec.m. , J Wsur Ec.m. , J 
Survival probability of the excited compound nucleus


 
Wsur E , J  Pxn E
*
CN
*
CN


n Ei* , J i

*
i 1  f Ei , J i
x


Total evaporation-residue cross section
J max
 ER Ec.m.    ER Ec.m. , J 
J 0
Maximal value of the angular momentum
x
Ec.m.  Q   Bn (i )   2 J max J max  1 2  0
i 1
Capture
Quantum diffusion approach
Sargsyan et al. EPJ A45, 125 (2010)
Sargsyan et al. EPJ A47, 38 (2011)
The quantum diffusion approach based on the following assumptions:
1.
2.
3.
The capture can be treated in term of a single collective variable: the
relative distance R between the colliding nuclei.
The internal excitations (for example low-lying collective modes such as
dynamical quadropole and octupole excitations of the target and
projectile, one particle excitations etc. ) can be presented as an
environment.
Collective motion is effectively coupled with internal excitations
through the environment.
Capture
The partial capture cross-section
 2
2 J  1 Pcap Ec.m. , J 
 cap Ec.m. , J  
2 Ec.m.
Pcap Ec.m. , J 
--- the partial capture probability at fixed energy and angular
momenta
The partial capture probability obtained by integrating the propagator G from the
initial state (R0,P0) at time t=0 to the finale state (R, P) at time t:
  Rm  Rt  
1
Pcap  lim  dR  dP G R, P, t R0 , P0 ,0  lim erfc 

t 
t  2


  RR t  
Rm

Dodonov and Man’ko, Trudy Fiz. Inst. Akad. Nauk 167, 7 (1986).
The quantum-mechanical dissipative effects and non-Markovian effects accompanying
the passage through the potential barrier are taken into consideration in our formalism.
Capture
Experimental data:
R. Bock et al., NPA 388, 334 (1982).
A. J. Pacheco et al., PRC 45, 2861 (1992).
E. Prokhorova et al., NPA 802, 45 (2008).
 The calculated capture cross section is in good agreement with the available
experimental data.
 With this approach, many heavy-ion capture reactions at energies above and well
below the Coulomb barrier have been successfully described.
Capture
219 MeV
215 MeV
 At small angular momenta, the
function σcap(Ec.m., J) growth is
due to the factor 2J + 1
 At larger J, the system of
colliding nuclei turns out in the
sub-barrier region and the
capture cross section falls down,
since in this region the decrease
of Pcap with J is not compensated
by the factor 2J + 1.
Complete fusion
In DNS model the compound nucleus is reached by a series of
transfers of nucleons or small clusters from the light nucleus to
the heavier one in a touching configuration.
Compound
Symmetric DNS
Quasifission
Complete fusion
The dynamics of DNC considered as an evolution of two coordinates:

A1  A2
A1  A2
-- the diffusion in mass asymmetry leads to fusion or symmetrization
R
-- the diffusion in the internuclear distance leads to quasifission
The probability of complete fusion

PCN 
  sym  R
We use two-dimensional Kramers-type expressions for the rates
Adamian at al., NPA 678, 24 (2000)
Adamian at al., NPA 633, 409 (1998)
In the reactions considered, the internal fusion barrier and quasifission barrier
sym
PCN  0.5
are almost equal to each other. Therefore,
and
  
  R
At treated narrow intervals of beam energies (about 28 MeV) and angular momenta (about
20), the dependence of PCN on Ec.m. and J can be neglected.
Channel widths and level density
The decay width of channel i is given in terms of the probability of this process
i 
RCN i

*
2  ECN
,J

The probability of neutron emission
RCN n
 ECN  Bn

*
*

ECN , J  
d  d ECN  Bn   , j TJ  A  1,  

 0

j  J 1/ 2



*
j  J 1/ 2


level density of the daughter nucleus

experimental values of neutron
binding energies are used:
Audi, NPA 729, 337 (2003)
transition coefficients: Mashnik, arXiv:nucl-th/0208048.
The fission probability in the case of a one-hump barrier


RCN f E , J 
*
CN

*
*
ECN
 B f ECN
,J

0

*
*
 f ECN
 B f ECN
, J   , J 
*
*
   d
1  exp 2   B f ECN
, J   ECN
The level density is calculated with the Fermi-gas model
Ignatyuk, Sov. J. Nucl. Phys. 29, 875 (1979).
Fission barrier
The fission barrier has the liquid-drop and microscopical parts


*
M


B f ECN
, J  B LD
J

B
f
f
-- liquid-drop part: Sierk, PRC 33, 2039 (1986)
B LD
f J 
B Mf  WsdA  WgrA
-- microscopical part: is related to the shell correction
at the ground state and the shell at the saddle point.
Usually, one neglects the shell correction at the saddle point



*
*
B Mf  B Mf ECN
 0  WgrA ECN
0

The predicted values of shell corrections:
Acta Phys. Pol. B 34, 2153 (2003), ibid. 34, 2073
(2003), ibid. 34, 2141 (2003); ibid. 32, 691 (2001)
The damping of the shell effects with the growth of the excitation energy and angular
momentum are approximated by the exponential damping factors






*
M
*
*


B f ECN
, J  B LD
J

B
E

0

exp

E
f
f
CN
CN J  ED  exp  J J  1 D
Fission barrier for 256No
The dependence of the liquid-drop and microscopical
parts of the fission barrier at E*CN=0.
 Solid, dashed, dash-dotted, and dotted curves correspond to
D = 600, 800, 1000, and 1200, respectively.
 While the value BfLD slowly decreases with the growth of J , the value BfM
rapidly falls down due to the damping of the shell effects.
 The difference between the microscopical components at the variation of
the parameter D is rather small at J < 10, but becomes considerable at
larger angular momenta.
without damping factor
The dependence of the total fission barrier at
E*CN=19.3 MeV and J = 0
Тhe dependence of the fission barrier on the angular
momentum is mainly governed by the damping of the
shell effects on the angular momentum!
Population of yrast rotational
states of 254No
Тhe relative transition intensities in the rotational band of 254No
E=215 MeV
E*CN=19.3 MeV
produced in the reaction
208Pb(48Ca,2n)254No
E=219 MeV
E*CN=22.7 MeV
Experimental data:
P. Reiter et al., PRL 82, 509 (1999)
P. Reiter et al., PRL 84, 3542 (2000).
Normalized to
transition 4+  2+
Normalized to
transition 8+  6+
 Higher beam energy leads to the large population of high-spin states. This difference is because
of the capture process: the partial capture cross section is larger at beam energy 219 MeV.
 The variation of the damping parameter D from 600 to 1200 does not strongly affect the results
of the calculation.
Entry spin distribution of
E=215 MeV
254No
Produced in the reaction 208Pb(48Ca,2n)254No
Calculations (lines) are done by using the damping
parameter D = 1000.
E=215 MeV
Experimental data:
P. Reiter et al., PRL 84, 3542 (2000).
 More high-spin states survive against
fission with increasing E. This could
be explained by a larger capture cross
section for large values of the initial
angular momentum at beam energy
219 MeV!
 Our results are close to the experimental
values, especially near the maxima of
spin distributions.
Evaporation-residue cross sections and
excitation functions of 48Ca+ 208Pb
Calculations (lines) are done by using the damping
parameter D = 1000.
Experimental data:
Belozerov et al., EPJ. A 16, 447 (2003)
Oganessian et al., PRC 64, 054606 (2001)
Gaggeler et al., NPA 502, 561 (1989)
Yeremin et al., JINR Rapid Commun. 6[92–98], 21 (1998).
 The systematic uncertainty in the definition of σER is
up to a factor of 3.
 The dependence of the calculated results on the
parameter D is rather weak. (in the maximum of a
2n evaporation channel, the variation of D from 600
to 1200 leads to the variation of σER from 2.5 to 3.6 μb.
Population of yrast rotational states of
and 250Fm
208Pb(48Ca,2n) 252No
204Hg(48Ca,2n) 250Fm
Experimental data:
Experimental data:
Herzberg et al., PRC 65, 014303 (2001).
Beam energy 215.5 MeV
Bastin et al., PRC 73, 024308 (2006).
Beam energy 209 MeV
exp
th
 ER
 220 nb,  ER
 210 nb
252No
exp
160
th
 ER
 980160
nb,  ER
 790 nb
Entry spin distribution of
252No
and 250Fm
The entry spin distributions of 250Fm and 252No,
48Ca(205 MeV)+204Hg→250Fm+2n (solid line),
48Ca(209 MeV)+204Hg→250Fm+2n (dashed line),
48Ca(211.5 MeV)+206Pb→252No+2n (dotted line),
48Ca(215.5 MeV)+206Pb→252No+2n (dash-dotted line).
Relative transition intensities in the yrast rotational
bands of 250Fm (solid line) and 252No (dotted line),
produced in the reactions
48Ca(205 MeV)+204Hg→250Fm+2n
48Ca(211.5 MeV)+206Pb→252No+2n.
Summary
 Using the statistical and quantum diffusion approaches, we studied the
population of rotational bands in superheavy nuclei produced in fusionevaporation reactions 206,208Pb(48Ca,2n)252,254No and 204Hg(48Ca,2n)250Fm.
 The Fermi-gas model was applied to calculate the level densities.
 The interval D = 600–1000 for the damping parameter was found for the
description of the damping of the shell effects with the angular momentum and
used in the calculations of the relative transition intensities in the ground-state
rotational bands, the entry spin distributions of the evaporation residues and
the evaporation-residue cross sections.
 Angular momentum dependence of these observables mainly comes from the
partial capture and survival probabilities.
 At low and moderate angular momentum values, the centrifugal forces
are not dangerous for the production of superheavy elements, especially in “the
island of stability.”